function [delta] = del_num( ln_m_h, ln_m, q, param)
    % solution for delta in replacement region given m_h and distribution of searchers q
    q_r = interp1( ln_m,q, ln_m_h, 'pchip');
    delta = nan(size(ln_m));
    tmp_ind = find(ln_m>ln_m_h,1);
    delta(1:tmp_ind-1) = param.s*param.lambda;
    delta(tmp_ind:end) = (param.zeta+param.s*param.lambda).*q_r./q(tmp_ind:end)-param.zeta;

end